Properties of nature that can be measured are known as quantitative properties, such as mass and length. A quantity is the amount of some property, such as the amount of mass or amount of length that an object has. Quantities can be measured in many different ways, so scientists have created standards so that everyone can know that they are talking about the same thing. For instance, length could be measured by the width of your finger, the length or your arm, or the distance that a horse can run in one day. Therefore, scientists decided to make the meter the standard unit of length to get rid of confusion. Scientists use units (such as meters) multiplied by numbers to express quantities (like “2.5 meters”). The standard system of measurement for scientists around the world and for most countries is known as the International System of Units, or SI.

4.2.1 SI units and some rules

There are seven base quantities, like length, corresponding to seven base units, like the meter (see Table 1). Units are the increments by which the quantities are measured. For instance, a quantity of length (l) is measured in units of meters (m).

Note that quantity symbols are written in italics whereas unit symbols are written upright. Unit symbols are considered mathematical entities and not abbreviations, so they are not followed by periods (except at the end of a sentence) or by an s for the plural.

The symbols for the ampere (A), Kelvin (K), and various other units are capitalized to show that they were named for a person. The unit spelled out is not capitalized so that the unit is not confused with the person for whom it is named.

Table 1: Base quantities and base units in the SI

Base Quantity

Quantity Symbol

Base Unit

Unit Symbol

length

l, h

meter

m

mass

m

kilogram

kg

time, duration

t

second

s

electric current

I

ampere

A

thermodynamic temperature

T

kelvin

K

amount of substance

n

mole

mol

luminous intensity

I_{V}

candela

cd

4.2.2 Derived units

Quantities defined in terms of the base quantities are known as derived quantities and are measured using derived units (see Appendix 8.1.0 for a table of derived quantities and units). For instance, velocity is measured as the amount of distance traveled per unit time, so it is derived from two base quantities (length and time). The unit symbol of velocity is therefore m/s.

4.2.3 Non-SI units

Some non-SI units are still widely used and usually defined in terms of SI units (see Appendix 8.2.0 for a table of non-SI units). The minute and liter are examples.

4.2.4 SI prefixes

SI prefixes are used to express the values of quantities either much larger than or much smaller than the SI unit without a prefix. Fourteen of the twenty SI prefixes are shown in Table 2.

Prefixes may be used with any SI base units or any derived units with special names (such as mm or kPa). Only one prefix may be used at a time.

Exponents. Any number to the power of 0 is 1. Therefore, 10^{0}= 1. See Table 2 for more decimal equivalents.

Positive exponents move the decimal to the right (10^{1} = 10. and 10^{6} = 1,000,000.). You can think of the exponent as specifying the number of zeros.

Negative exponents move the decimal to the left (10^{-1} = 0.1 and 10^{-6} = 0.000001). If you write the number properly with a zero before the decimal, then you can still think of the exponent as specifying the number of zeros.

Table 2: SI prefixes

Prefix

Symbol

Exponential Multiplier

Decimal Equivalent

peta

P

10^{15}

1 000 000 000 000 000.

tera

T

10^{12}

1 000 000 000 000.

giga

G

10^{9}

1 000 000 000.

mega

M

10^{6}

1 000 000.

kilo

k

10^{3}

1 000.

hecto

h

10^{2}

100.

deca

da

10^{1}

10.

10^{0}

1.

deci

d

10^{-1}

0.1

centi

c

10^{-2}

0.01

milli

m

10^{-3}

0.001

micro

μ

10^{-6}

0.000 001

nano

n

10^{-9}

0.000 000 001

pico

p

10^{-12}

0.000 000 000 001

femto

f

10^{-15}

0.000 000 000 000 001

4.2.5 Scientific Notation

Scientists use scientific notation to make it easier to write very large or very small numbers.

Form. Scientific notation takes the following form

a × 10^{b}

where

a = any number

b = an integer (whole number)

The value of b determines how many times the decimal was moved to write the original number in scientific notation. The decimal is moved so that there is only one nonzero digit to the left of it.

Examples.

Example 1: The mass of a proton is about 0.000 000 000 000 000 000 000 000 001 672 621 71 kg. In scientific notation, this number is written as 1.672 621 71 × 10^{-27} kg.

Example 2: The mass of the earth is about 5 973 600 000 000 000 000 000 000 kg. In scientific notation, this number is written as 5.973 6 × 10^{24} kg.

To write a number in scientific notation, move the decimal so that there is only one nonzero digit to the left of it, and count how many spaces it was moved. The number of spaces moved will become the exponent. Remove the place-holding zeros like in the examples above.

To write small numbers, move the decimal to the right; the resulting exponent will be negative. In Example 1 above, the decimal had to be moved 27 times to the right so that there would be only one nonzero digit to the left of it.

To write large numbers, move the decimal to the left; the resulting exponent will be positive. In Example 2 above, the decimal had to be moved 24 times to the left so that there would be only one nonzero digit to the right of it.

Sometimes you will have to add, subtract, multiply, or divide numbers in scientific notation (which happens to be much easier than writing out 25 zeros!). Table 3 explains the rules for making calculations with numbers written in scientific notation. After finishing a calculation, always adjust the value to proper scientific notation if necessary and pay attention to significant figures.

Table 3: Rules for Calculations with Numbers in Scientific Notation

Rule

Example

Addition and Subtraction 1. Change one of the values so that it has the same exponent as the other value. 2. Add or subtract the first factors and multiply by the factor of 10 obtained in step 1.

Problem: 5.9 × 10^{3} – 3.3 × 10^{2} 1. 5.9 × 10^{3} can be written as 59 × 10^{2} so the problem becomes 59 × 10^{2} – 3.3 × 10^{2} 2. 59 × 10^{2} – 3.3 × 10^{2} = 55.7 × 10^{2} = 5.6 × 10^{3}

Multiplication 1. Multiply the first factors of the number and add the exponents.

Converting SI units from one to another is as simple as multiplying or dividing by factors of ten, which amounts to moving decimal points (see Figure 2).

The exponent of the units can be used to find the direction and number of times to move the decimal point. To do this, determine the exponent of the starting unit and then subtract from it the exponent of the desired unit. Here are a couple examples:

Example 1: Convert 2.5 km to cm. The exponent for km is 3, and the exponent for cm is -2. Subtract the two numbers: 3 – (-2) = 5. Therefore, move the decimal 5 places to the right, giving 250,000 cm.

Example 2: Convert 2.5 cm to km. The exponent for cm is -2, and the exponent for km is 3. Subtract the two numbers: -2 – 3 = -5. Therefore, move the decimal 5 places to the left, giving 0.000025 km.

Figure 2 This chart shows how to convert a unit with one prefix to a unit with a different prefix.

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VCS Science Handbook Section 4.2

4.2.0 Units of measurementquantitative properties, such as mass and length. Aquantityis the amount of some property, such as the amount of mass or amount of length that an object has. Quantities can be measured in many different ways, so scientists have created standards so that everyone can know that they are talking about the same thing. For instance, length could be measured by the width of your finger, the length or your arm, or the distance that a horse can run in one day. Therefore, scientists decided to make the meter the standard unit of length to get rid of confusion. Scientists useunits(such as meters) multiplied bynumbersto express quantities (like “2.5 meters”). The standard system of measurement for scientists around the world and for most countries is known as the International System of Units, orSI.l) is measured in units of meters (m).italicswhereas unit symbols are written upright. Unit symbols are considered mathematical entities and not abbreviations, so they are not followed by periods (except at the end of a sentence) or by an s for the plural.l,hmtITnI_{V}derivedquantities and are measured using derived units (see Appendix 8.1.0 for a table of derived quantities and units). For instance, velocity is measured as the amount of distance traveled per unit time, so it is derived from two base quantities (length and time). The unit symbol of velocity is therefore m/s.Exponents. Any number to the power of 0 is 1. Therefore, 10^{0}= 1. See Table 2 for more decimal equivalents.^{1}= 10. and 10^{6}= 1,000,000.). You can think of the exponent as specifying the number of zeros.^{-1}= 0.1 and 10^{-6}= 0.000001). If you write the number properly with azero before the decimal, then you can still think of the exponent as specifying the number of zeros.^{15}^{12}^{9}^{6}^{3}^{2}^{1}^{0}^{-1}^{-2}^{-3}^{-6}^{-9}^{-12}^{-15}Form. Scientific notation takes the following form^{b}Examples.Example 1:The mass of a proton is about 0.000 000 000 000 000 000 000 000 001 672 621 71 kg. In scientific notation, this number is written as 1.672 621 71 × 10^{-27}kg.Example 2:The mass of the earth is about 5 973 600 000 000 000 000 000 000 kg. In scientific notation, this number is written as 5.973 6 × 10^{24}kg.only one nonzero digit to the left of it, and count how many spaces it was moved. The number of spaces moved will become the exponent. Remove the place-holding zeros like in the examples above.write small numbers, move the decimal to the right; the resulting exponent will be negative. In Example 1 above, the decimal had to be moved 27 times to the right so that there would be only one nonzero digit to the left of it.write large numbers, move the decimal to the left; the resulting exponent will be positive. In Example 2 above, the decimal had to be moved 24 times to the left so that there would be only one nonzero digit to the right of it.Addition and Subtraction1.Change one of the values so that it has the same exponent as the other value.2.Add or subtract the first factors and multiply by the factor of 10 obtained in step 1.Problem:5.9 × 10^{3}– 3.3 × 10^{2}1.5.9 × 10^{3}can be written as 59 × 10^{2}so the problem becomes 59 × 10^{2}– 3.3 × 10^{2}2.59 × 10^{2}– 3.3 × 10^{2}= 55.7 × 10^{2}= 5.6 × 10^{3}Multiplication1.Multiply the first factors of the number and add the exponents.Problem:(5.9 × 10^{3})(3.3 × 10^{2})1.(5.9 × 3.3) × 10^{(3+2)}= 19.47 × 10^{5}= 1.9 × 10^{6}Division1.Divide the first factors of the number and subtract the exponent in the denominator from the exponent in the numerator.Problem:(5.9 × 10^{3})/(3.3 × 10^{2})1.(5.9/3.3) × 10^{(3-2)}= 1.7879 × 10^{1}= 18Example 1:Convert 2.5 km to cm. The exponent for km is 3, and the exponent for cm is -2. Subtract the two numbers: 3 – (-2) = 5. Therefore, move the decimal 5 places to the right, giving 250,000 cm.Example 2:Convert 2.5 cm to km. The exponent for cm is -2, and the exponent for km is 3. Subtract the two numbers: -2 – 3 = -5. Therefore, move the decimal 5 places to the left, giving 0.000025 km.Figure 2 This chart shows how to convert a unit with one prefix to a unit with a different prefix.